- 7.1.1: Show that y 2 3ex 1 e22x is a solution of the differential equation...
- 7.1.2: Verify that y 2t cos t 2 t is a solution of the initial-value probl...
- 7.1.3: Show that y e2at cos t is a solution of the differential equation y...
- 7.1.4: (a) Show that every member of the family of functions y sln x 1 Cdy...
- 7.1.5: (a) What can you say about a solution of the equation y9 2y 2 just ...
- 7.1.6: a) What can you say about the graph of a solution of the equation y...
- 7.1.7: ogistic growth A population is modeled by the differential equation...
- 7.1.8: The Fitzhugh-Nagumo model for the electrical impulse in a neuron st...
- 7.1.9: Explain why the functions with the given graphs cant be solutions o...
- 7.1.10: The function with the given graph is a solution of one of the follo...
- 7.1.11: Match the differential equations with the solution graphs labeled I...
- 7.1.12: Von Bertalanffys equation states that the rate of growth in length ...
- 7.1.13: 1315 Drug dissolution Differential equations have been used extensi...
- 7.1.14: 1315 Drug dissolution Differential equations have been used extensi...
- 7.1.15: 1315 Drug dissolution Differential equations have been used extensi...
- 7.1.16: The logistic differential equation Suppose that the per capita grow...
- 7.1.17: Modeling yeast populations Use the fact that the per capita growth ...
- 7.1.18: Modeling yeast populations (cont.) Verify that Nstd 42e 0.55t 209.8...
Solutions for Chapter 7.1: Modeling with Differential Equations
Full solutions for Biocalculus: Calculus for Life Sciences | 1st Edition
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
A symmetric matrix with eigenvalues of both signs (+ and - ).
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Solvable system Ax = b.
The right side b is in the column space of A.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).